Q.A rectangular parallelepiped (box) is drawn with its edges parallel to the coordinate axes, one vertex at the origin $O(0,0,0)$ and the diagonally opposite vertex at $P(2,4,5)$; the three edges from $O$ run along the $x$-, $y$- and $z$-axes. Let $F$ be the vertex of this box that is the foot of the perpendicular dropped from $P$ onto the XZ-plane (the plane $y=0$) — i.e. the corner having the same $x$- and $z$-coordinates as $P$ but lying in the plane $y=0$. Find the coordinates of $F$.

3D Coordinate Geometry

You already know 2D coordinate geometry — the $xy$-plane where every point is described by two numbers $(x, y)$. Now imagine lifting that plane into the air. That is three-dimensional geometry.

The Intuition: Three Numbers, One Point

In the real world you rarely locate something with just two numbers. To describe where a book sits on a shelf you might say: "third shelf up, fourth book from the left, and it is the one nearest the wall." That is three pieces of information — height, sideways position, and depth.

In 3D coordinate geometry we do exactly this. We keep the familiar $x$ and $y$ axes (which define a flat floor) and add a third axis — the $z$-axis — pointing straight up. Every point in space now needs three numbers: $(x, y, z)$.

The Precise Statement

Definition: A rectangular 3D coordinate system consists of three mutually perpendicular number lines — the $x$-axis, $y$-axis and $z$-axis — meeting at a common point, the origin $O(0, 0, 0)$. Any point $P$ in space is uniquely represented by an ordered triple $(x, y, z)$, where:

• $x$ = signed distance from the $yz$-plane,
• $y$ = signed distance from the $zx$-plane,
• $z$ = signed distance from the $xy$-plane.

How to Read a 3D Point

Take the point $A(2, -3, 4)$. Start at the origin. Move 2 units along the $x$-axis. From there move $-3$ units parallel to the $y$-axis (backward, because it is negative). From that spot move 4 units parallel to the $z$-axis (upward). You have reached $A$.

The Three Coordinate Planes

Each pair of axes determines a plane:

These three planes cut space into 8 octants (the 3D analogue of the four quadrants of the plane). The first octant is where $x > 0$, $y > 0$ and $z > 0$.

Distance Between Two Points

This is the natural extension of the 2D distance formula. For $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$:

It is just the diagonal of a rectangular box whose edges are the differences in $x$, $y$ and $z$. The distance of $P$ from the origin is the special case $OP = \sqrt{x_1^2 + y_1^2 + z_1^2}$.

Section Formula (Internal Division)

If $R$ divides the segment joining $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ internally in the ratio $m : n$, then:

This is the same pattern as the 2D section formula, applied to three coordinates instead of two.

What Comes Next

Once you are comfortable with points, distances and section division, the natural next steps (in later study) are direction cosines and the equations of lines and planes in space. For now, remember the core idea: 3D coordinate geometry is 2D geometry with one extra dimension — every formula you already know simply gains a third term.

3D Coordinate Geometry is the heart of the NCERT Class 11 Mathematics chapter Introduction to Three Dimensional Geometry, matching searches such as "3D coordinate geometry formulas class 11 maths" or "distance and section formula in 3D important questions". The octants, coordinate planes, distance formula and section formula introduced here carry real weightage in CBSE Class 11 exams and form the groundwork for the Class 12 three-dimensional geometry of lines and planes, as well as the coordinate-geometry sections of JEE Main and state CETs.